JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer The real values of \[x\]and\[y\]for which the equation is \[(x+iy)\] \[(2-3i)\]= \[4+i\] is satisfied, are [Roorkee 1978]

    A) \[x=\frac{5}{13},y=\frac{8}{13}\]

    B) \[x=\frac{8}{13},y=\frac{5}{13}\]

    C) \[x=\frac{5}{13},y=\frac{14}{13}\]

    D) None of these

    Correct Answer: C

    Solution :

      Equation \[(x+iy)(2-3i)=4+i\] Þ  \[(2x+3y)+i(-3x+2y)=4+i\] Equating real and imaginary parts, we get \[2x+3y=4\]    ......(i) \[-3x+2y=1\] ......(ii) From (i) and (ii), we get \[x=\frac{5}{13},y=\frac{14}{13}\] Aliter: \[x+iy=\frac{4+i}{2-3i}=\frac{(4+i)(2+3i)}{13}=\frac{5}{13}+\frac{14}{13}i\] .

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